If the server isn't showing up on the kids' computers, but it shows the green light indicating that it is, indeed, up and running - then it leads me to think they may not be on the same network. Do the kids' computer have ip addresses within the same range? Can they ping the server computer? I have also run into the issue, when using LAN cables, that if I don't disable the wireless the computer will try to function on wired and wireless simultaneously and that can really mess things up.
Otherwise, maybe the problem lies in the router. I had this problem during one camp I ran where everything was set up perfect - but wouldn't work. After an hour I just reset the router to factory defaults and it worked perfectly afterwards.
Just some thoughts - I hope it works out for you!